Indexes can be used as investment tools in various ways. For instance, indexes comprising a plurality of securities can often be bought and sold more cheaply than buying and selling the individual constituents of the index. This allows investment in these securities with reduced transaction costs. In passive and enhanced indexing, investments are made with reference to an index or benchmark portfolio. A benchmark portfolio is a portfolio intended to represent the market in general. The holdings of a benchmark portfolio are often proportional to the market capitalization of each security. Performance statistics such as return and risk are reported with respect to the reference index or benchmark portfolio. Indexes can serve as active manager benchmarks or the underlyers for investable products such as ETFs and mutual funds.
Factor indexes are indexes designed to replicate the returns associated with a selected quantitative factor. Possible factors include those present in commercial risk models. These include style factors, industry factors, market factors, country factors, and currency factors. Axioma's U.S. Equity Fundamental Factor Risk Model uses ten style factors: exchange rate sensitivity, growth, leverage, liquidity, market sensitivity, medium term momentum, short term momentum, size, value, and volatility.
U.S. Pat. No. 7,698,202 describes characteristics, properties and uses of factor risk models. This patent is incorporated by reference herein in its entirety.
Factors are defined quantitatively. For instance, the ten factors in Axioma's U.S. Equity Fundamental Factor Risk Model are defined as follows.
The exchange rate sensitivity factor measures the sensitivity of a stock's historical returns to the returns of a currency basket, referenced in U.S. dollars. The basket used is the International Monetary Fund's Special Drawing Rights (SDR) which contains the currencies U.S. dollar, euro, Japanese yen, and pound sterling. A stock's exchange rate sensitivity factor exposure is the normalized slope obtained by regressing the time series of its past 120 day return against that of the currency basket.
The growth factor gives an indication of a company's historical rate of growth. The growth factor is calculated as the product of the one year return on equity times one minus the dividend payout rate. The return on equity is calculated as the ratio of the annualized income over the last year to the common equity value of a year ago. The dividend payout rate is calculated as the ratio of the annualized dividends per share to the annualized earnings per share. The annualized income over the last year is calculated from income before extraordinary income analogously to dividends per share and earnings per share by going to the most recent filing in the last 520 trading days and summing the four most recently filed values over the previous 15 months. The common equity value of a year ago is taken as the most recent value for common equity that has been filed in the period between 780 and 260 trading days ago, which is approximately three years to one year ago.
The leverage factor provides a measure of a company's exposure to debt levels. The leverage factor is calculated as total debt divided by market capitalization. Total debt is the sum of long term debt and debt in current liabilities, short term debt, taken on the most recent date over the last 520 trading days for which both values have been filed. For market capitalization, the 20 day average is used.
The liquidity factor provides a measure of a stock's trading activity, or lack thereof. It is defined as the natural logarithm of the last 20 day average volume divided by the natural logarithm of the last 20 day average market capitalization, both expressed in currency values such as dollars.
The market sensitivity factor is a measure of a stock's under or over performance relative to the broad market from historical data. It is simply a stock's historical beta calculated by regressing the time series of an asset's returns against the market returns over the preceding 120 trading days. The beta is then adjusted for autocorrelation in the asset returns and asynchronous trading via the Scholes-Williams procedure with a lag/lead of one.
The medium term momentum factor gives a measure of a stock's past performance over the medium term time horizon. It is defined as an asset's cumulative return over the last 250 trading days, excluding the last 20 trading days.
The short term momentum factor gives a measure of a stock's recent performance. It is defined as an asset's cumulative return over the last 20 trading days.
The size factor differentiates between large and small stocks and is defined as the natural logarithm of the market capitalization, averaged over the last 20 trading days. Market capitalization is computed as the product of the total shares outstanding and closing price.
The value factor gives a measure of how fairly a stock is priced within the market. The value factor is calculated as the ratio of common equity to the current market capitalization (i.e. Book-to-Price). The calculation uses the common equity value reported on the most recent date from the last 520 trading days, which is approximately two years. Market capitalization is taken as shares outstanding times the closing price on the day for which the risk model is generated.
The volatility factor gives a measure of an asset's relative volatility over time according to its historical behavior. It is calculated as the square root of the asset's absolute return averaged over the last 60 days, divided by the cross-sectional standard deviation of the risk model's estimation universe.
Axioma's U.S. Equity Statistical Factor Risk Model uses a different set of factors to describe the universe of assets. These factors can be used to define a factor. A factor may be defined by one risk model, while risk and active risk, which is also known as tracking error, is predicted with a different risk model.
There are numerous other possible factors. U.S. Pat. No. 7,620,577 lists a number of these: market price, market capitalization, book value, sales, revenue, earnings, earnings per share, income, income growth rate, dividends, dividends per share, earnings before interest, tax, depreciation and amortization, and the like. This patent is incorporated by reference herein in its entirety.
For many factors, there are many similar methods to quantitatively measure and define the factor score or factor exposure of an asset to the factor. For example, value factors commonly use several similar metrics including the book-to-price ratio, the earnings-to-price ratio, and the earnings-per-share ratios. In fact, Axioma's software is capable of expressing a fully specified risk model in terms of slightly different factors while still giving good risk estimates. Factor indexes constructed using similar factor scores would be similar even if the factor scores were slightly different.
In factor indexes, the goal is to create a set of holdings whose returns replicate the returns associated with one or more factors. The return of a single factor can be defined in numerous ways. It can refer to the return of a long-short, dollar neutral portfolio that is long in those assets with the highest factor scores and short in those assets with the lowest factors scores. The weights can cover a fixed percentage of the assets in the set of securities, measured in terms of capitalization or equal weighting. The weights used for each security can be equally weighted or cap-weighted. Alternatively, the return of a factor can be derived from a regression. This regression can be a cross sectional regression that regresses across assets or a times series regression that regresses over time. Numerous other methods have been proposed for defining the return of a factor, such as the return of a factor mimicking portfolio.
Often an additional desirable characteristic for a factor index is to make the returns of the factor index dissimilar to the returns associated with non-targeted factors. Dissimilar can mean a number of different things. It can refer to uncorrelated returns, or, it can refer to limiting or neutralizing the exposure to a factor, for example. When a factor index is dissimilar to non-targeted factors, it is described as being pure.
The exposure of a portfolio to a factor is often measured with respect to a benchmark portfolio. The exposure of the benchmark portfolio is the sum of the products of the factor scores for each asset times the weight of that asset in the benchmark portfolio. Since this is the exposure of the benchmark, this exposure value is representative of the market in general. The exposure of the factor index is the sum of the products of the factor scores for each asset times the weight of that asset in the factor index. The difference in these two exposures is termed the active exposure. That is, the active exposure is the exposure of the factor index minus the exposure of the benchmark. Small positive or negative active exposures indicate that the factor index is similar to the benchmark. Large positive or large negative active exposures indicate that the factor index is dissimilar to the benchmark. For a pure factor index, the active exposure to the targeted factor is substantially different than zero, while the active exposure of the factor index to non-targeted factors is small.
Exposures and active exposures are often expressed as a Z score, which measures the exposure in terms of standard deviations. An active exposure Z score of 100% indicates that the factor index exposure is one standard deviation greater than the benchmark's exposure. Such a large active exposure would indicate that the exposure of the factor index and the benchmark portfolio are substantially different. A Z score of −100% indicates that the factor index exposure is one standard deviation less than the benchmark's exposure and would also indicate a substantial difference between the factor index and benchmark portfolio exposures. On the other hand, an active exposure Z score between −5% and 5% would indicate that the difference in exposures between the factor index and the benchmark portfolio is less than one twentieth of one standard deviation and would indicate that the two sets of holdings have similar exposures to the factor.
It will be recognized that what constitutes a substantial difference in exposures between the benchmark portfolio and the factor index is a subjective determination by each investor. Whereas some investors may consider a Z score difference of 25% to be small, others may believe that 25% represents a substantial difference. The same may be true for Z score differences of 20%, 30%, 40%, or even 50%.
Factor indexes described in the prior art suffer a number of deficiencies.
Consider first simple factor indexes. Simple factor indexes are sets of holdings created by selecting a universe of potential investments, defining a factor score for each element or asset in the universe, ranking the factor scores, and then buying those assets with the highest scores and selling those assets with the lowest scores. Buying refers to making a long position and selling refers to making a short positions. In some factor indexes, only the highest scores are bought, or, alternatively, only the lowest scores are bought instead of sold. These two alternatives produce long only indexes. The number of assets bought and sold can be determined in many different ways. The number can be related to the market capitalization of the assets or it can be a fixed number. The amount bought or sold in each asset can also be determined in a number of ways including cap-weighting or equal weighting. The simple factor index may or may not be dollar neutral, meaning that the total amount of assets bought equals the total amount of assets sold. As one particular example, consider a 35% cap-weighted, long-short dollar neutral. This simple factor index is created by buying the 35% of a universe measured by market capitalization with the highest factor scores, and selling the 35% of the universe with the lowest scores. The individual asset weights are proportional to market capitalization.
Simple factor indexes have a number of undesirable properties. The turnover and the number of names involved are generally large. This can render investing in the index expensive and impractical. In addition, the resulting portfolios are often not neutral to other factors such as value, leverage, and even size. The returns associated with these simple indexes are not pure returns, but also comprise returns associated with non-vanishing factor exposures. Investment in these simple factor indexes can therefore easily involve unintended bets on other factors.
Another kind of factor index is a factor mimicking portfolio constructed from a matrix of factor exposures. See for example, R. Litterman, Modern Investment Management: An Equilibrium Approach, John Wiley and Sons, Inc., Hoboken, N.J., 2003 (Litterman), which gives detailed descriptions of factor mimicking portfolios and which is incorporated by reference herein in its entirety. Often, the matrix of factor exposures is associated with a factor risk model, but that need not be the case. Factor mimicking portfolios are designed so that they have exposure to one and only one factor. The exposure to all other factors in the risk model or matrix of factor exposures is zero by construction. Factor mimicking portfolios are pure.
Unfortunately, factor mimicking portfolios are expensive to buy and trade. Furthermore, there is no explicit control over the returns associated with these portfolios. In practice, the returns of a factor mimicking portfolio may be quite different than the returns associated with a factor.
A number of new products have been introduced that are similar to simple factor indexes. These new factor index products all explicitly control implementation costs such as turnover, the number of names held, and the like. However, none of these new products use a tracking error constraint or minimize tracking error when constructing the products. Instead, they control or maximize the exposure to the targeted factor.
Factor index products that construct factor indexes by controlling the factor exposure of the target exposure are much more likely to underperform compared to those products that explicitly constrain the tracking error of the factor index. That is, the returns of factor indexes using exposure control are likely to differ from the true factor returns, defined here as the returns of a target factor portfolio.